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# Mathematical Economics Question, specific issues in the annex....

Mathematical Economics Question, specific issues in the annex.

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Question 1 (12 points) A. Find the total diferential in each o± the ±ollowing cases: (4 points) y=(x+z)(x-2z) where x=2-7z y= ax 2 + bxz + cu, where x=eu+hv and z=gu where a,b,c w, h and g are constants. Then ²nd the total derivative dy/dz in the ²rst equation, and the partial total derivatives ∂y/∂u and ∂y/∂v in the second. B. In each o± the ±ollowing cases, ²nd (dy/dx) using the implicit-±unction rule. (4 points) F(y,x) = 3x 2 + 2xy + 4y 3 = 0 F(y,x) = 12x 5 -2y= 0 F(y,x) = xy 2 e y C. For each o± the ±ollowing equations F(y, x)=0, is an implicit ±unction y=±(x) de²ned around point x=1 and y=3? (4 points) x 3 -2x 2 y +3xy 2 -22 =0 2x 2 + 4xy -y 4 + 67=0 Question 2 (8 points) Consider the ±ollowing model o± the market in which the government taxes the production o± the good. Q d = D(P) Demand ±unction Q s = S(P^,w o ) Supply ±unction Q s = Q d Equilibrium condition P= P^ +t o The ±ollowing restrictions hold: D P < 0, S P^ > 0, S t < 0, S w < 0 Here Q stands ±or quantity demanded/supplied, P ±or the price paid by consumers, P^ ±or the price received by producers, t is the tax per unit on the good and w is the given wage o± workers used in the production o± this good. Both wages and the tax are exogenous. a. First eliminate P^ ±rom the model by substituting ±or P^ in the supply ±unction. Then use the equilibrium condition to reduce the model to one equation, and write it in the ±orm: F(endogenous variable, exogenous variables) =0. (2 points) b. Veri±y whether the implicit ±unction theorem holds. (2 points) c. Then determine the impact on equilibrium price P o± a change in (i) wages and (ii) taxes using the implicit ±unction rule. (4 points)
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